let S, T be RealNormSpace; :: thesis: for I being LinearOperator of S,T
for Z being Subset of S st I is one-to-one & I is onto & I is isometric holds
( Z is closed iff I .: Z is closed )
let I be LinearOperator of S,T; :: thesis: for Z being Subset of S st I is one-to-one & I is onto & I is isometric holds
( Z is closed iff I .: Z is closed )
let Z be Subset of S; :: thesis: ( I is one-to-one & I is onto & I is isometric implies ( Z is closed iff I .: Z is closed ) )
assume that
A1: ( I is one-to-one & I is onto ) and
A2: I is isometric ; :: thesis: ( Z is closed iff I .: Z is closed )
P1: dom I = the carrier of S by FUNCT_2:def 1;
consider J being LinearOperator of T,S such that
P2: ( J = I " & J is one-to-one & J is onto & J is isometric ) by A1, A2, LM020;
thus ( Z is closed iff I .: Z is closed ) :: thesis: verum

proof

thus ( Z is closed implies I .: Z is closed ) by A1, A2, LM023; :: thesis: ( I .: Z is closed implies Z is closed )
assume P3: I .: Z is closed ; :: thesis: Z is closed
J .: (I .: Z) = I " (I .: Z) by A1, P2, FUNCT_1:85
.= Z by FUNCT_1:94, P1, A1 ;
hence Z is closed by P2, P3, LM023; :: thesis: verum

end;